Inner Model Theory

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The fundamental open problem of inner model theory is to extend the theory to models satisfying stronger large cardinal hypotheses, with ``There is a supercompact cardinal" being the most well-known target. This target has resisted 40 years of effort by set theorists, despite major advances in the late 60's, in 1974-1980, and in 1985-1994. That earlier work gave us a good understanding of canonical inner models satisfying ``There is a Woodin cardinal", and even slightly stronger hypotheses, but it fell well short of superstrong cardinals. Since 1994, progress in this interval has been slow and incremental, but Steel believes it has opened up a promising line of attack. This line builds on concepts and methods derived from the evolving merger of inner model theory and pure descriptive set theory. It centers on the Core Model Induction technique, and the Mouse Set Conjecture. Steel proposes to work in this direction. Work on the set-theoretic foundations of mathematics has revealed a remarkable phenomenon. It appears that all natural mathematical theories fall into a single hierarchy, calibrated by the degree to which a commitment to the existence of large infinite sets is implicit in the theory. Moreover, this degree of commitment corresponds exactly to the power of the theory to decide questions about concrete objects, like natural numbers, real numbers, or sets of real numbers. Set theorists have developed two complementary methods for locating theories in this hierarchy of logical strength, namely Forcing and Inner Model Theory. While the basics of Forcing are understood in full generality, the basics of Inner Model Theory are not, and consequently many problems concerning the logical strengths of particular foundational theories remain open. Steel will work to further develop the basics of Inner Model Theory.